It seems to me the main difference between your output and mine is that your output looks like it worked, and mine failed. After all, I'm asking for help because it doesn't work.

Well, I think there really ought to be some output in there. But since you say it doesnt, and it looks kinda like what you get in Re^2: LWP Can't POST to HTTPS, I think we've reached the end of our investigation

I think its time to go to rt.cpan.org and report this to maintainers of LWP-Protocol-https/IO-Socket-SSL/Net-SSLeay

At the very least there should be more debugging output with a more useful error message , something like you get with openssl client